The left above panel of Fig. REF , shows the evolution of the structure growth of the matter, \(\mathrm {f\sigma _{8}}\) , along
\(\mathrm {z}\)  for the different cosmologies. These curves are comparable with each other at \(\mathrm {z<0.4}\)  but they deviate one after another at
\(\mathrm {z>0.4}\) . It implies that they are sensitive to the background cosmology. Within the matter era the amplitude of \(\mathrm {f\sigma _{8}}\)  in the
IDE model is enhanced relative to the \(\omega \) DE model at \(\mathrm {z}<3\) , but both are smaller than that found in the \(\Lambda \) CDM. At \(\mathrm {z}>5\) ,
\(\bar{Q}\)  and \(\mathrm {\omega _{DE}}\)  would brough about a large structure formation in the IDE and \(\omega \) DE scenarios, respectively. Due to the fact that
the amount of DM is bigger than the amount of DE at earlier times; therefore, it produces an enhancement on the amplitudes of \(\mathrm {f\sigma _{8}}\) 
in the IDE model respect to that found in the \(\Lambda \) CDM, respectively. Our fitting results are consistent at \(1\sigma \)  error with those reported by
[1]}, [2]}, [3]}, [4]}, [5]}.
The right above panel of this Figure depicts the evolution of the total matter power for different scenarios at \(\mathrm {z=1.6}\) . Notice that \(\mathrm {P(k)}\)  in the
IDE model is enhanced with respect to that in the \(\Lambda \) CDM but it is suppressed in relation to that found in the \(\omega \) DE scenario. That could be
a consequence of the amount of concentrated matter at early times and also the presence of \(\bar{Q}\) . Moreover, the vertical line indicates the turnover
position, \(\mathrm {k_{eq}}\) , is very close in our models. Also, we notice a series of wiggles on the \(\mathrm {P(k)}\)  due
to the coupling between the photons and baryons before recombination; namely, the presence of baryons have left their effect there. these
arguments, the IDE and \(\omega \) DE can be distinguished from the curves of \(\mathrm {f\sigma _{8}}\) , and the structure growth data could break the
possible degeneracy between these two models and provides a signature to discriminate them. These outcomes are in correspondence with those found
by [2]}, [4]}, [5]} at \(1\sigma \)  error.
Likewise, the below panel of this Figure displays the effects of \(\bar{Q}\)  and \(\mathrm {\omega _{DE}}\)  on the amplitude of the CMB temperature power
spectrum at low multipoles \(\textit {l}<100\) , in where the amplitude of the integrated Sachs-Wolfe effect is deviated in the IDE model respect to that
found in the other scenarios. Instead, at high multipoles \({\textit {l}>100}\) , \(\bar{Q}\)  and \(\mathrm {\omega _{DE}}\)  increase the concentration of DM
early times, affecting the sound horizon at the end, which shiftes to right the values of the acoustic peaks located at
\(\textit {l}=n_{p}\pi D_{A}(z_{*})/r_{s}(z_{*}),\, n_{p}=1,2,3..\) , and reduced the amplitudes of the first peaks when the studied models are compared.
These features can be understood by considering the extra-terms proportionals to \(\mathrm {{\lambda }_{0}}\) , \(\mathrm {{\lambda }_{1}}\)  and
\(\mathrm {{\lambda }_{2}}\)  in the \(\mathrm {\Omega _{DM}}\) , which increases and, in consequence, amplifies the amount of DM at early times. That is in accordance with the result found in the
previous panel of this Figure and with those found in [2]}, [4]}, [5]} at \(1\sigma \)  error.
